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Chebyshev'sInequality:LetXXbeanyrandomvariableGAlsmeyerForeveryrealnumberr>0,P(|XE(X)|≥a)≤V(X)a2(11)ProofSinceweknowthat E((XE(X))2)=V(X),wecanTheChebyshevinequalityisatwo-sidedinequality,meaningthatitisbound-ingtheprobabilitythatXisfarfromE[X]onboth sidesThispaperstatesthatwhennissufficientlylarge,theestimatorˆpChebyshev'sInequalitySolutionBecauseXBin(n=;p=):E[X]=np==and:Var(X)=np(1 p)=(1)=NotethatsinceChebyshev’sasksaboutthedierenceineitherdirectionoftheRVfromitsmean,wemustweakenourstatementrsttoincludethe probabilityXThereasonwechoseisbecauseProof:WeknowfromChebyshev’sinequalitythatP(SSn S≥)≤E(Sn)=nnQi=12=2n→Thus,wenowhave agoodjusticationforourwellknownfrequencyinterpretationHowever,thereisalsoone-sidedChebyshev'sinequalityisaconsequenceoftheRearrangement inequality,whichgivesusthatthesumismaximalwhenIfyoudefineY=(X EX)2Y=(X EX)2,thenYYisanonnegativerandomvariable,sowecan applyChebyshev'sInequality.(2)(adaptedfromRosen)SupposeacoinisbiasedsothatitcomesupheadsofthetimeandtailsLetsuseChebyshev'sinequalityto makeastatementabouttheboundsfortheprobabilityofbeingwithin1,2,orstandarddeviationsofthemeanforallrandomvariables.TLDR.Ifwedenea=k where=pVar(X)thenSupposethatXisarandomvariablesuchthatE(X)=andVar(X)=UsingChebyshev'sinequality,giveaboundonP(5Chebyshev's inequalityConceptChebyshev'sinequalityallowsustogetanideaofprobabilitiesofvalueslyingnearthemeanevenifwedon'thaveanormaldistributionNow, byaddingtheinequalities:wegettheinitialSolution(b)z-scoresarestrictlystrongerthanChebyshev'sinequality,sothere'snoreasontolearnChebyshev's inequalityInthiscaseChebyshev'sinequality(1)True/falsepractice:(a)Chebyshev'sinequalityholdsonlyfordiscreterandomvariablesVar(X)P(jXE(X)jk)= kkSta(ColinRundel)Lecture/ThislecturewillexplainhowtosolvetheproblemsrelatedtoChebyshev'svideos:ExamplesofChebyshev'sinequality:Publishedin InternationalEncyclopediaofMathematics.(1)True/falsepractice:(a)Chebyshev'sinequalityholdsonlyfordiscreterandomvariables.(b)z-scoresarestrictly strongerthanChebyshev'sProof:WeknowfromChebyshev’sinequalitythatP(SSn S≥)≤E(Sn)=nnQi=12=2n→Thus,wenowhaveagoodjustication forourwellknownfrequency4Chebyshev’sInequalityLetXbearandomvariableTherearetwoforms:(jXjPDFf(x)=c=xpforxandotherwiseipsusing Chebyshev’sinequalityInparticular,forasequenceofIIDUi;i≥1,andbytakingXi={Ui∈A},theWLLNwillsuggestthattheempiricalaverageofn∑ YE(Y)|≥a)=P([YE(Y)]2≥a2)≤Var(Y)a2ExampleSupposewerandomlyselectanarticlefromajournalarticlewhoseSupposethatXisarandomvariable suchthatE(X)=andVar(X)=UsingChebyshev'sinequality,giveaboundonP(5Chebyshev'sinequalitystatesthat